In this paper, we consider the problem of private information retrieval with secure storage (SS-PIR) under a setting with disjoint colluding sets, where the user wishes to privately retrieve one out ofKindependent messages that are securely stored acrossNservers. TheNservers are partitioned intoMdisjoint colluding sets, i.e., within them-th group,m∈[M], any set of up toTmcolluding servers cannot learn any information about the index of the desired message, and any set of up toXmcolluding servers cannot learn any information about theKmessages. The asymptotic capacity is defined as the maximum possible number ofq-ary symbols of the desired message that can be retrieved perq-ary downloaded symbol, in the limit as the number of messagesK→ ∞. We demonstrate that the asymptotic capacity of SS-PIR with disjoint colluding sets is the solution to a linear program parameterized by the server partition, privacy thresholdsTm, and security thresholdsXm. Our achievability scheme introduces a novel pre-decoding strategy built upon cross-subspace alignment (CSA) codes. In this strategy, instead of requiring the user to decode the message from all of the original CSA coded answer symbols, our approach allows certain servers to perform a local pre-decoding and return intermediate results that turn out to be more communication-efficient. This mechanism is the key to minimizing the download cost, allowing our scheme to match the information-theoretic converse and thereby establish the asymptotic capacity of SS-PIR with disjoint colluding sets.

The problem of weakly-private information retrieval (WPIR) is to allow the user to retrieve one out of$K$messages from a set of$N$distributed servers while guaranteeing that the information about the identity of the index of the desired message is leaked to the servers in a controlled manner, i.e., the privacy level cannot be less than a prescribed threshold as measured by a privacy metric. In this work, we consider the converse-induced privacy metric (CIPM) for WPIR proposed by Jia and fully settle the capacity, i.e., the supremum of the average number of desired message symbols retrieved per downloaded symbol, for the arbitrary number of messages$K$and the number of servers$N$. Our achievability scheme makes use of a symmetrized (with respect to the servers) version of the PIR codes due to Tian et al. by carefully designing the query distributions. It turns out that the optimal trade-off between the privacy level and the reciprocal of the retrieval rate in terms of the CIPM metric is linear for non-degenerate settings.

We consider the problem ofX-secure andT-private linear computation with graph based replicated storage (GXSTPLC), which enables the user to privately retrieve a linear combination of messages from a set ofNdistributed servers where each message is restricted to be stored exclusively among a subset of servers, adhering to anX-security constraint. This constraint dictates that any group of up toXcolluding servers must not disclose any information about the stored messages. Furthermore, any group of up toTservers is restricted from learning anything about the coefficients of the linear combination retrieved by the user. In this work, we completely characterize the asymptotic capacity of GXSTPLC, i.e., the supremum of achievable rates (which is the average number of desired symbols retrieved per downloaded symbol), in the limit as the number of messagesKapproaches infinity. Specifically, it is shown that a prior linear programming based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is tight (thus settles their conjecture) by constructing achievability schemes. Notably, our achievability scheme also settles the exact capacity (i.e., for finiteK) ofX-secure linear combination with graph based replicated storage (GXSLC). Our achievability proof builds upon an achievability scheme for a closely related problem named asymmetric X-secure T-private linear computation with graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform security and privacy levels across messages and coefficients (of the desired linear combination). In particular, by carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity achieving schemes for GXSTPLC. In regard to the achievability scheme for Asymm-GXSTPLC, interesting aspects of our construction include a novel query and answer design which makes use of a Vandermonde decomposition of Cauchy matrices, and a trade-off among message replication, security and privacy thresholds.

The problem of X-secure T-private linear computation with graph based replicated storage (GXSTPLC) is to enable the user to retrieve a linear combination of messages privately from a set of N distributed servers where every message is only allowed to replicate among a subset of servers subject to an X-security constraint, i.e., any groups of up to X colluding servers must reveal nothing about the messages. Besides, any groups of up to T servers must reveal no information about the coefficients of the linear combination retrieved by the user. In this paper, inspired by a Vandermonde decomposition of Cauchy matrices, we propose an achievability scheme for GXSTPLC that achieves the rate of (ρmin−X −T)/N if every message is replicated at least ρmintimes and ρmin> X + T, which coincides with a lower bound of the rate of X-secure T-private information retrieval with graph based replicated storage (GXSTPIR) by Jia and Jafar. Moreover, the asymptotic capacity of GXSTPLC is partially settled, including the setting where the storage forms a symmetric pattern.

Sarcopenia is correlated with poor clinical outcomes in breast cancer (BC) patients. However, there is no precise quantitative study on the correlation between body composition changes and BC metastasis and survival. The present study proposed a deep learning radiomics (DLR) approach to investigate the effects of muscle and fat on distant metastasis and death outcomes in BC patients. Image feature extraction was performed on 4th thoracic vertebra (T4) and 11th thoracic vertebra (T11) on computed tomography (CT) image levels by DLR, and image features were combined with clinical information to predict distant metastasis in BC patients. Clinical information combined with DLR significantly predicted distant metastasis in BC patients. In the test cohort, the area under the curve of model performance on clinical information combined with DLR was 0.960 (95% CI: 0.942-0.979, P < 0.001). The patients with distant metastases had a lower pectoral muscle index in T4 (PMI/T4) than in patients without metastases. PMI/T4 and visceral fat tissue area in T11 (VFA/T11) were independent prognostic factors for the overall survival in BC patients. The pectoralis muscle area in T4 (PMA/T4) and PMI/T4 is an independent prognostic factor for distant metastasis-free survival in BC patients. The current study further confirmed that muscle/fat of T4 and T11 levels have a significant effect on the distant metastasis of BC. Appending the network features of T4 and T11 to the model significantly enhances the prediction performance of distant metastasis of BC, providing a valuable biomarker for the early treatment of BC patients.